Institute of Space Sciences Zeta Functions, vacuum fluctuations and Cosmology Emilio Elizalde www.ice.csic.es/personal/elizalde/eli/eli.htm QUANTUM FIELD THEORY AND GRAVITY (QFTG’2018) International Conference, TSPU, Tomsk, Russia Jul 30 - Aug 5, 2018 Name or Title or Xtra Zero point energy QFT vacuum to vacuum transition: 0 H 0 h | | i Spectrum, normal ordering (harm oscill): 1 † H = n + λn an an 2 ~ c 1 0 H 0 = λn = tr H h | | i 2 Xn 2 gives physical meaning? ∞ Regularization + Renormalization ( cut-off, dim, ζ ) Even then: Has the final value real sense ? Institute of Space Sciences Effects of the Quantum Vacuum a) Negligible: Sonoluminiscence, Schwinger ̴10-5 b) Important : Wetting He3 – alcali ̴30% c) Incredibly big: Cosmological constant ̴10120 Name or Title or Xtra Riemann Zeta Function s a.c. 1 ζ (s) = Σ −− s n ζ(s) pole −4 −3 −2 −1 0 1 2 3 4 5 Institute of Space Sciences Name or Title or Xtra F Yndurain, A Slavnov "As everybody knows ..." Institute of Space Sciences = / 2 − 2 1 = Re = 1 − 2 = + 1 1 Real on the critical line + = Z(t) 2(−) 푠 2 푠 1 −푖 Hardy, Riemann-Siegel Name or Title or2 Xtra Operator Zeta F’s in MΦ: Origins The Riemann zeta function ζ(s) is a function of a complex variable, s. To define it, one ∞ starts with the infinite series 1 ns nX=1 which converges for all complex values of s with real Re s > 1, and then defines ζ(s) as the analytic continuation, to the whole complex s plane, of the function given, Re s > 1, − by the sum of the preceding series. Leonhard Euler already considered the above series in 1740, but for positive integer values of s, and later Chebyshev extended the definition to Re s > 1. Godfrey H Hardy and John E Littlewood, “Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Math 41, 119 (1916) Did much of the earlier work, by establishing the convergence and equivalence of series regularized with the heat kernel and zeta function regularization methods G H Hardy, Divergent Series (Clarendon Press, Oxford, 1949) Srinivasa I Ramanujan had found for himself the functional equation of the zeta function Torsten Carleman, “Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes" (French), 8. Skand Mat-Kongr, 34-44 (1935) Zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold for the case of a compact region of the plane Robert T Seeley, “Complex powers of an elliptic operator. 1967 Singular Integrals" (Proc. Sympos. Pure Math., Chicago, Ill., 1966) pp. 288-307, Amer. Math. Soc., Providence, R.I. Extended this to elliptic pseudo-differential operators A on compact Riemannian manifolds. So for such operators one can define the determinant using zeta function regularization D B Ray, Isadore M Singer, “R-torsion and the Laplacian on Riemannian manifolds", Advances in Math 7, 145 (1971) Used this to define the determinant of a positive self-adjoint operator A (the Laplacian of a Riemannian manifold in their application) with eigenvalues a1, a2, ...., and in this case the zeta function is formally the trace −s ζA(s) = Tr (A) the method defines the possibly divergent infinite product ∞ a = exp[ ζ ′(0)] n − A nY=1 J. Stuart Dowker, Raymond Critchley “Effective Lagrangian and energy-momentum tensor in de Sitter space", Phys. Rev. D13, 3224 (1976) Abstract The effective Lagrangian and vacuum energy-momentum tensor < T µν > due to a scalar field in a de Sitter space background are calculated using the dimensional-regularization method. For generality the scalar field equation is chosen in the form (2 + ξR + m2)ϕ = 0. If ξ = 1/6 and m = 0, the µν µν 2 4 1 renormalized < T > equals g (960π a )− , where a is the radius of de Sitter space. More formally, a general zeta-function method is developed. It yields the renormalized effective Lagrangian as the derivative of the zeta function on the curved space. This method is shown to be virtually identical to a method of dimensional regularization applicable to any Riemann space. Q Stephen W Hawking, “Zeta function regularization of path integrals in curved spacetime", Commun Math Phys 55, 133 (1977) This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises to n dimensions by adding extra flat dims. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fifth dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This EM tensor has an anomalous trace. Basic strategies Jacobi’s identity for the θ function − n2 iπτ θ (z, τ) := 1 + 2 ∞ q cos(2nz), q := e , τ C 3 n=1 ∈ P2 1 z /iπτ z 1 θ3(z, τ) = e θ3 − equivalently: √ iτ τ | τ − ∞ 2 ∞ 2 2 (n+z) t π π n e− = e− t cos(2πnz), z, t C, Ret > 0 r t ∈ nX= Xn=0 −∞ Higher dimensions: Poisson summ formula (Riemann) f(~n) = f(m~ ) ~nXZp m~XZp ∈ ∈ e f Fourier transform [Gelbart + Miller, BAMS ’03, Iwaniec, Morgan,e ICM ’06] Truncated sums asymptotic series −→ 2nd Reading December 22, 2014 15:18 WSPC/S0219-8878 IJGMMP-J043 1550019 International Journal of Geometric Methods in Modern Physics Vol. 12 (2015) 1550019 (28 pages) c World Scientific Publishing Company DOI: 10.1142/S021988781550019X Zeta functions on tori using contour integration Emilio Elizalde Institute of Space Science, National Higher Research Council ICE–CSIC, Facultat de Ciencies, Campus UAB Torre C5-Par-2a, 08193 Bellaterra (Barcelona), Spain [email protected] [email protected] Klaus Kirsten GCAP–CASPER, Department of Mathematics Baylor University, One Bear Place, Waco TX 76798, USA klaus [email protected] Nicolas Robles∗ Institut f¨ur Mathematik, Universit¨at Z¨urich Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland [email protected] Floyd Williams Department of Mathematics and Statistics, Lederle Graduate Research Tower, 710 North Pleasant Street University of Massachusetts, Amherst MA 01003-9305, USA [email protected] Received 26 August 2013 by Prof. Dr. Emilio Elizalde on 01/05/15. For personal use only. Accepted 29 October 2014 Published 24 December 2014 Int. J. Geom. Methods Mod. Phys. Downloaded from www.worldscientific.com A new, seemingly useful presentation of zeta functions on complex tori is derived by using contour integration. It is shown to agree with the one obtained by using the Chowla– Selberg series formula, for which an alternative proof is thereby given. In addition, a new proof of the functional determinant on the torus results, which does not use the Kronecker first limit formula nor the functional equation of the non-holomorphic Eisenstein series. As a bonus, several identities involving the Dedekind eta function are obtained as well. Keywords: Tori; zeta functions; spectral theory; functional determinants. Mathematics Subject Classification 2010: Primary 11M41, 81T40; Secondary 81T25 ∗Corresponding author. 1550019-1 The Chowla-Selberg Formula (CS) M. Lerch, Sur quelques formules relatives du nombre des classes, Bull. Sci. Math. 21 (1897) 290-304 A. Selberg and S. Chowla, On Epstein’s Zeta function (I), Proc. Nat. Acad. Sci. 35 (1949) 371-74 S. Chowla and A. Selberg, On Epstein’s Zeta function, J. reine angew. Math. (Crelle’s J.) 227 (1967) 86-110 K. Ramachandra, Some applications of Kronecker’s limit formulas, Ann. Math. 80 (1964) 104-148 A. Weil, Elliptic functions according to Eisenstein and Kronecker (Springer, Berlin, 1976) S. Iyanaga and Y. Kawada, Eds., Encyclopedic Dictionary of Mathematics, Vol. II (The MIT Press, Cambridge, 1977), pp. 1378-79 B.H. Gross, On the periods of abelian integrals and a formula of Chowla and Selberg, Inv. Math. 45 (1978) 193-211 P. Deligne, Valeurs de fonctions L et periodes d’integrales, PSPM 33 (1979) 313-346 History Lerch (1897): D | | D λ h log Γ = h log D log(2π) log a λ D | | − 3 − Xλ=1 (a,b,cX) 2 + log θ′ (0 α)θ′ (0 β) 3 1 | 1 | D discriminant, θ η3 (a,b,cX) 1′ ∼ h class number of binary quadratic forms (a, b, c) Eta evaluations Dedekind eta function for Im (τ) > 0 1/24 n 2πiτ η(τ) = q ∞ (1 q ), q := e n=1 − It is a 24-th root of the discriminantQ func ∆(τ) of an elliptic curve C/L from a lattice L = aτ + b a, b Z { | ∈ } ∞ ∆(τ) = (2π)12q (1 qn)24 Langlands program: − geom of number th nY=1 Extended CS Series Formulas (ECS) Consider the zeta function (Res > p/2, A > 0, Req > 0) −s ′ 1 ′ ζ (s) = (~n + ~c)T A (~n + ~c) + q = [Q (~n + ~c) + q]−s A,~c,q 2 ~nX∈Zp ~nX∈Zp prime: point ~n = ~0 to be excluded from the sum (inescapable condition when c1 = = cp = q = 0) ··· Q+ (~n + ~c) q = Q(~n) + L(~n) +q ¯ Case q = 0 (Req > 0) 6 p/2 p/2 s s/2+p/4+2 s s/2+p/4 (2π) q − Γ(s p/2) 2 π q− ζA,~c,q(s) = − + √det A Γ(s) √det A Γ(s) T 1 s/2 p/4 T 1 ′ cos(2π ~m ~c) ~m A− ~m − Kp/2 s 2π 2q ~m A− ~m − × Zp · ~mX1 2 p ∈ / [ECS1] Pole: s = p/2 Residue: (2π)p/2 Res ζ (s) = (det A)−1/2 s=p/2 A,~c,q Γ(p/2) Gives (analytic cont of) multidimensional zeta function in terms of an exponentially convergent multiseries, valid in the whole complex plane Exhibits singularities (simple poles) of the meromorphic continuation —with the corresponding residua— explicitly Only condition on matrix A: corresponds to (non negative) quadratic form, Q.